Official OS X Mountain Lion.app Installer from the Mac App Store.
Both versions 10.8.4 AND 10.8.5 full installers included (zipped).
Un-archive
the zip archive and you will get the app Installer. A bootable ISO or
bootable USB can be made directly from the InstallESD.dmg disk image found under Contents/SharedSupport/. Instructions on how to do
that are on the Internet/Youtube.
https://osxdaily.com/2012/02/17/make-bootable-os-x-10-8-mountain-lion-usb-install-drive/
NOTE: If you get a damaged message when launching, no need to worry. Run this command in Terminal:
xattr -cr "wherever your installer is. drag and drop into terminal, and remove quotes."
OR
xattr -c "wherever your installer is. drag and drop into terminal, and remove quotes."
NOTE:
While installing, if you get an error message such as Apple can't
verify your computer, change the date setting October 1st, 2012 in
Terminal or System Preferences. It should work after that.
The checksums for the zip file and the InstallESD.dmg disk image are listed in the CHECKSUM_INFO.txt